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Let's assume that there is a linear formula that gives us P(n) using P(k) where 0 <= k <= n-1. P(n) is the number of positions at depth n.

We have then the following formula:

P(n) = sum(R(k)*P(n-k)) for 1 <= k <= n

Calculating R(k) using rokiki's results we deduce :

 1 12                   = 12*1                   
 2 114                  = 12*12                  -30*1                   
 3 1068                 = 12*114                 -30*12                  +60*1                   
 4 10011                = 12*1068                -30*114                 +60*12                  -105*1